Nevertheless, it is conjectured that, for a variety \(X\)  over a finitely generated field \(k\)  of characteristic zero with \(X(k)\)  Zariski-dense, the only obstruction to the Hilbert property should indeed come from unramified covers (see Conjecture REF  below for a precise statement). Further taking into account that the original version of Hilbert's irreducibility theorem does not need to contend with any unramified covers at all (since \(\mathbb {P}_n\)  is geometrically simply connected), it seems natural to rule out such covers at the outset. This leads to the following notion introduced in [1]}:
